Answer:
a)
[tex]y=400(2.5)^{x}[/tex]
b)
3,814,698
c)
16.08 weeks
Step-by-step explanation:
a)
The question presented here is similar to a compound interest problem. We are informed that there are 400 rice weevils at the beginning of the study. In a compound interest problem this value would be our Principal.
P = 400
Moreover, the population is expected to grow at a rate of 150% every week. This is equivalent to a rate of interest in a compound interest problem.
r = 150% = 1.5
The compound interest formula is given as;
[tex]A=P(1+r)^{n}[/tex]
We let y be the weevil population in any given week x. The formula that can be used to predict the weevil population is thus;
[tex]y=400(1+1.5)^{x}\\\\y=400(2.5)^{x}[/tex]
b)
The weevil population 10 weeks after the beginning of the study is simply the value of y when x = 10. We substitute x with 10 in the equation obtained from a) above;
[tex]y=400(2.5)^{10}\\\\y=3814697.3[/tex]
Therefore, the weevil population 10 weeks after the beginning of the study is approximately 3,814,698
c)
We are simply required to determine the value of x when y is
1,000,000,000
Substitute y with 1,000,000,000 in the equation obtained in a) above and solve for x;
[tex]1000000000=400(2.5)^{x}\\\\2.5^{x}=2500000\\\\xln(2.5)=ln(2500000)\\\\x=\frac{ln(2500000}{ln(2.5)}=16.0776[/tex]
What is the volume of the right triangular prism in cubic meters?
Answer:
Volume of the Right Triangular Prism is 1771 m³.
Step-by-step explanation:
Given:
A Right Triangular base Prism.
Length of the legs of the right triangle of the base is 14 m , 23 m
Hypotenuse of the triangle is 26.9 m
Height of the Prism is 11 m
To find: volume of the Prism.
We know that Volume of the Prism = Base Area × Height
Volume of the Right Triangular Prism = Area of Base Triangle × Height
= 1/2 × 14 × 23 × 11
= 7 × 23 × 11
= 1771 m³
Therefore, Volume of the Right Triangular Prism is 1771 m³.
Answer:
1,771
Step-by-step explanation:
Write the ordered pair that represents yz. Then find the magnitude of yz . y(-2,5),z(1,3)
ANSWER
[tex]|^{ \to} _{YZ}| = \sqrt{13} [/tex]
EXPLANATION
Given the points, y(-2,5),z(1,3)
[tex] ^{ \to} _{YZ} = \binom{1}{3} - \binom{ - 2}{5} = \binom{3}{ - 2} [/tex]
Therefore the ordered pair is <3,-2>
The magnitude is
[tex] |^{ \to} _{YZ}| = \sqrt{ {3}^{2} + ( - 2)^{2} } [/tex]
[tex] |^{ \to} _{YZ}| = \sqrt{ 9 +4} [/tex]
[tex]|^{ \to} _{YZ}| = \sqrt{13} [/tex]
Answer: AAAAAAAAAAAAAAAAAAAAAAAAAAa
Convert 88 square yards to square meters (to the nearest tenth).
To convert 88 square yards to square meters, we use the conversion factor 1 square yard = 1.196 square meters. By multiplying 88 by 1.196, we find that 88 square yards is approximately 105.3 square meters.
Explanation:The question is part of the mathematics subject, specifically in the area of unit conversion. We have a conversion factor to use, which is 1 square yard = 1.196 square meters, based on the provided reference information.
So to convert 88 square yards to square meters, you multiply 88 by 1.196.
88 yards2 * 1.196 m2/yard2 = 105.3 m2.
so 88 square yards is approximately = 105.3 square meters.
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find 2(cos 240+isin 240) ^4 (answer choices below)
1. C. -512√3+512i
2. B. 16(cos240°+i sin240°)
3. D. 3√2+3√6i, -3√2-3√6i
4. A. cos60°+i sin60°, cos180°+i sin180°, cos300°+i sin300°
5. D. 2√3(cos π/6+i sin π/6), 2√3(cos 7π/6+i sin 7π/6)
We will see that the equivalent expression is:
[tex]8*(cos(240\°) + i*sin(240\°))[/tex]
So the correct option is the first one.
How to rewrite the given expression?
We have the expression:
[2*(cos(240°) + i*sin(240°))]^4
Remember that Euler's formula says that:
[tex]e^{ix} = cos(x) + i*sin(x)[/tex]
Then we can rewrite our expression as:
[tex][2*(cos(240\°) + i*sin(240\°)]^4 = [2*e^{i*240\°}]^4[/tex]
Now we distribute the exponent:
[tex]2^4*e^{4*i*240\°} = 8*e^{i*960\°}[/tex]
Now, we need to find an angle equivalent to 960°.
Remember that the period of the trigonometric functions is 360°, then we can rewrite:
960° - 2*360° = 240°
This means that 960° is equivalent to 240°. Then we can write:
[tex]8*e^{i*960\°} = 8*e^{i*240\°} = 8*(cos(240\°) + i*sin(240\°))[/tex]
So the correct option is the first one.
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A football stadium has an attendance of 4997 people. Of these, 2118 are cheering for Team A and 2568 are female. Of the people cheering for Team A, 982 are female. Find the probability that a randomly selected attendee is female or cheers for Team A. (a) Are the events "cheering for Team A" and "being a female" mutually exclusive? No Yes (b) What is the probability that a randomly selected attendee is female or cheers for Team A? nothing (Type an integer or decimal rounded to three decimal places as needed.)
The events “cheering for Team A” and “being a female” are not mutually exclusive. The probability that a randomly selected attendee is female or cheers for Team A is approximately 0.741.
Explanation:(a) No, the events “cheering for Team A” and “being a female” are not mutually exclusive. This is because there are females who are cheering for Team A. Mutually exclusive events cannot happen at the same time.
(b) To find the probability that a randomly selected attendee is female or cheers for Team A, we need to add the probabilities of each event happening and subtract the probability of both events happening at the same time. We can use the formula:
P(A or B) = P(A) + P(B) - P(A and B)
In this case, P(A) is the probability of cheering for Team A, P(B) is the probability of being female, and P(A and B) is the probability of being a female who cheers for Team A.
Given the numbers provided, the probability of cheering for Team A is 2118/4997 and the probability of being female is 2568/4997. The probability of being a female who cheers for Team A is 982/4997. Plugging these values into the formula, we get:
P(Female or Team A) = P(Team A) + P(Female) - P(Female and Team A) = 2118/4997 + 2568/4997 - 982/4997 = 3704/4997 ≈ 0.741
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The pep squad sold c, cheeseburgers and h, hothogs at the friday night football game. A total of 220 were sold. There were 3 times more hotdogs sold than cheeseburgers. Write a system of equations for this situation.
Answer:
c + h = 220h = 3cStep-by-step explanation:
The total sold is the sum of the individual numbers sold, hence c+h.
We assume "3 times more" means "3 times as many", so the number of hotdogs sold (h) is 3 times the number of cheeseburgers sold (c), hence 3c.
c + h = 220
h = 3c
_____
55 cheeseburgers and 165 hotdogs were sold.
Which of the following statements reflects the principles of avoiding distractions and being other-oriented?
a.
“Let’s go outside. It’s really noisy in here and I can’t really hear what you are saying.”
b.
“I’ll just get this call and then we can talk.”
c.
“I think you should just quit. There’s no sense in being miserable, I always say.”
d.
None of the above
Answer: A
Step-by-step explanation:
Answer:
a.
“Let’s go outside. It’s really noisy in here and I can’t really hear what you are saying.”
Step-by-step explanation:
Which of the following statements reflects the principles of avoiding distractions and being other-oriented?
a. “Let’s go outside. It’s really noisy in here and I can’t really hear what you are saying.”
This is clear from option A - the person is saying that its noisy here and he cannot listen to the other person.
Sketch the graph of y=2(x-2)2+5 and identify the axis of symmetry.
Answer:
x=2
Step-by-step explanation:
Use the quadratic formula to solve the equation.
4x^2– 10x + 5 - 0
Enter your answer in simplified radical form
X=_____ X=_____
Answer:
[tex]\large\boxed{x=\dfrac{5-\sqrt5}{4},\ x=\dfrac{5+\sqrt5}{4}}[/tex]
Step-by-step explanation:
[tex]\text{The quadratic formula for}\ ax^2+bx+c=0\\\\\text{if}\ b^2-4ac<0,\ \text{then the equation has no real solution}\\\\\text{if}\ b^2-4ac=0,\ \text{then the equation has one solution:}\ x=\dfrac{-b}{2a}\\\\\text{if}\ b^2-4ac,\ ,\ \text{then the equation has two solutions:}\ x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\==========================================[/tex]
[tex]\text{We have the equation:}\ 4x^2-10x+5=0\\\\a=4,\ b=-10,\ c=5\\\\b^2-4ac=(-10)^2-4(4)(5)=100-80=20>0\\\\x=\dfrac{-(-10)\pm\sqrt{20}}{2(4)}=\dfrac{10\pm\sqrt{4\cdot5}}{8}=\dfrac{10\pm\sqrt4\cdot\sqrt5}{8}=\dfrac{10\pm2\sqrt5}{8}\\\\=\dfrac{2(5\pm\sqrt5)}{8}=\dfrac{5\pm\sqrt5}{4}[/tex]
Grading Scale 1 has the following weights- (Tests- 50% Quiz- 25% Homework- 15% Final Exam- 10%). Calculate your final average if your performance in the class is as follows- Test Grades- 78 Quiz- 87 Homework- 80 Final Exam- 90.
a. 78(50) + 87(25) + 80(15) + 90(10) = X
b. 78(.50) + 87(.25) + 80(.15) + 90(.10) = X
c. 78(5) + 87(2.5) + 80(1.5) + 90(.10) = X
d. 78(50) + 87(.25) + 80(1.5) + 90(.10) = X
Answer:
78(0.50)+87(0.25)+80(0.15)+90(0.10)=x
Step-by-step explanation:
Let
x-----> the final average
we know that
To find the final average multiply each performance by its weight in decimal and then sum the results
x=78(0.50)+87(0.25)+80(0.15)+90(0.10)
x=39+21.75+12+9
x=81.75
An ice cream store offers a bowl with one giant scoop or two
regular scoops of ice cream for $2.75. A giant scoop is a sphere with a diameter of 6 centimeters. A regular scoop is a
sphere with a diameter of 4 centimeters. Which is closest to
the greatest volume of ice cream that can be purchased for $2.75?
A 67 cm
B 113 cm
C 536
D 905 cm
Answer:
B
Step-by-step explanation:
The volume of a sphere is given by
[tex]V=\frac{4}{3}\pi r^3[/tex]
where r is the radius
For $2.75, we can get 1 large OR 2 small scoops.
Giant scoop has diameter 6, so radius is half of that, which is 3, hence the volume is:
[tex]V=\frac{4}{3}\pi r^3\\V=\frac{4}{3}\pi (3)^3\\V=113.1[/tex]
Regular scoop's diameter is 4, hence radius is 2. So volume of 1 regular scoop is:
[tex]V=\frac{4}{3}\pi r^3\\V=\frac{4}{3}\pi (2)^3\\V=33.51[/tex]
We can get 2 of those, so total volume is 33.51 + 33.51 = 67.02
Hence, the max volume for $2.75 is around 113, answer choice B.
The water tank in the diagram is in the shape of an inverted right circular cone. The radius of its base is 16 feet, and its height is 96 feet. What is the height, in feet, of the water in the tank if the amount of water is 25% of the tank’s capacity?
Answer:
6433.98 ft
Step-by-step explanation:
In order to find what 25% of the tank's capacity is, we know to know the full capacity of the tank then take 25% of that. The volume formula for a right circular cone is
[tex]V=\frac{1}{3}\pi r^2h[/tex]
We have all the values we need for that:
[tex]V=\frac{1}{3}\pi (16)^2(96)[/tex]
This gives us a volume of 25735.93 cubic feet total.
25% of that:
.25 × 25735.93 = 6433.98 ft
Answer:
The height of the water is [tex]60.5\ ft[/tex]
Step-by-step explanation:
step 1
Find the volume of the tank
The volume of the inverted right circular cone is equal to
[tex]V=\frac{1}{3}\pi r^{2} h[/tex]
we have
[tex]r=16\ ft[/tex]
[tex]h=96\ ft[/tex]
substitute
[tex]V=\frac{1}{3}\pi (16)^{2} (96)[/tex]
[tex]V=8,192\pi\ ft^{3}[/tex]
step 2
Find the 25% of the tank’s capacity
[tex]V=(0.25)*8,192\pi=2,048\pi\ ft^{3}[/tex]
step 3
Find the height, of the water in the tank
Let
h ----> the height of the water
we know that
If two figures are similar, then the ratio of its corresponding sides is proportional
[tex]\frac{R}{H}=\frac{r}{h}[/tex]
substitute
[tex]\frac{16}{96}=\frac{r}{h}\\ \\r= \frac{h}{6}[/tex]
where
r is the radius of the smaller cone of the figure
h is the height of the smaller cone of the figure
R is the radius of the circular base of tank
H is the height of the tank
we have
[tex]V=2,048\pi\ ft^{3}[/tex] -----> volume of the smaller cone
substitute
[tex]2,048\pi=\frac{1}{3}\pi (\frac{h}{6})^{2}h[/tex]
Simplify
[tex]221,184=h^{3}[/tex]
[tex]h=60.5\ ft[/tex]
How do you simplify this expression step by step?
[tex]\bf \textit{Pythagorean Identities} \\\\ sin^2(\theta)+cos^2(\theta)=1\implies cos^2(\theta)=1-sin^2(\theta) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{csc(\theta )-sin(\theta )}{cos(\theta )}\implies \cfrac{~~\frac{1}{sin(\theta )}-sin(\theta )~~}{cos(\theta )}\implies \cfrac{~~\frac{1-sin^2(\theta )}{sin(\theta )}~~}{cos(\theta )}[/tex]
[tex]\bf \cfrac{1-sin^2(\theta )}{sin(\theta )}\cdot \cfrac{1}{cos(\theta )}\implies \cfrac{\stackrel{cos(\theta )}{\begin{matrix} cos^2(\theta ) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}} }{sin(\theta )}\cdot \cfrac{1}{\begin{matrix} cos(\theta ) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix} }\implies \cfrac{cos(\theta )}{sin(\theta )}\implies cot(\theta )[/tex]
Answer:
cot Ф
Step-by-step explanation:
Recall that sin²Ф + cos²Ф = 1, (which also says that cos²Ф - 1 = sin²Ф).
Also recall the definitions of the csc, sin and cos functions.
Your expression is equivalent to:
1 sin Ф
---------- - -------------
sin Ф 1
===================
cos Ф
There are three terms in your expression: csc, sin and cos. Multiply all of them by sin Ф. The result should be:
1 - sin²Ф
---------------
sin Ф · cos Ф
Using the Pythagorean identity (see above), this simplifies to
cos²Ф
------------------
sin Ф·cos Ф
and this whole fraction reduces to
cos Ф
-------------- and this ratio is the definition of the cot function.
sin Ф
Thus, the original expression is equivalent to cot Ф
Use the quadratic formula to solve the equation.
4x^2 - 10x + 5 = 0
Enter your answers, in simplified radical form.
X=_____ or X=_____
ANSWER
[tex]x = \frac{ 5 - \sqrt{ 5} }{4} \: or \: \: x = \frac{ 5 + \sqrt{ 5} }{4} [/tex]
EXPLANATION
The given quadratic equation is
[tex]4 {x}^{2} - 10x + 5 = 0[/tex]
We compare this to
[tex]a {x}^{2} + bx + c = 0[/tex]
to get a=4, b=-10, and c=5.
The quadratic formula is given by
[tex]x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} [/tex]
We substitute these values into the formula to get:
[tex]x = \frac{ - - 10 \pm \sqrt{ {( - 10)}^{2} - 4(4)(5)} }{2(4)} [/tex]
This implies that
[tex]x = \frac{ 10 \pm \sqrt{ 100 - 80} }{8} [/tex]
[tex]x = \frac{ 10 \pm \sqrt{ 20} }{8} [/tex]
[tex]x = \frac{ 10 \pm2 \sqrt{ 5} }{8} [/tex]
[tex]x = \frac{ 5 \pm \sqrt{ 5} }{4} [/tex]
The solutions are:
[tex]x = \frac{ 5 - \sqrt{ 5} }{4} \: or \: \: x = \frac{ 5 + \sqrt{ 5} }{4} [/tex]
Answer:
[tex]\large\boxed{x=\dfrac{5-\sqrt5}{4},\ x=\dfrac{5+\sqrt5}{4}}[/tex]
Step-by-step explanation:
[tex]\text{The quadratic formula for}\ ax^2+bx+c=0\\\\\text{if}\ b^2-4ac<0,\ \text{then the equation has no real solution}\\\\\text{if}\ b^2-4ac=0,\ \text{then the equation has one solution:}\ x=\dfrac{-b}{2a}\\\\\text{if}\ b^2-4ac,\ ,\ \text{then the equation has two solutions:}\ x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\==========================================[/tex]
[tex]\text{We have the equation:}\ 4x^2-10x+5=0\\\\a=4,\ b=-10,\ c=5\\\\b^2-4ac=(-10)^2-4(4)(5)=100-80=20>0\\\\x=\dfrac{-(-10)\pm\sqrt{20}}{2(4)}=\dfrac{10\pm\sqrt{4\cdot5}}{8}=\dfrac{10\pm\sqrt4\cdot\sqrt5}{8}=\dfrac{10\pm2\sqrt5}{8}\\\\=\dfrac{2(5\pm\sqrt5)}{8}=\dfrac{5\pm\sqrt5}{4}[/tex]
what property does the following expression demonstrate 9(3x)=27(x)
Answer:
Associative property of multiplication
Step-by-step explanation:
To show that 9(3x) = 27(x), we need to show that 9(3x) = (9 * 3)x.
The APM does just that. By this property, in multiplication, the order of which numbers are multiplied do not matter.
So, 9(3x) = (9 * 3)x.
And by multiplication, (9 * 3)x = 27x.
So 9(3x) = 27x
Verify that the given differential equation is not exact. (−xy sin(x) + 2y cos(x)) dx + 2x cos(x) dy = 0 If the given DE is written in the form M(x, y) dx + N(x, y) dy = 0, one has My = Nx = . Since My and Nx equal, the equation is not exact. Multiply the given differential equation by the integrating factor μ(x, y) = xy and verify that the new equation is exact. If the new DE is written in the form M(x, y) dx + N(x, y) dy = 0, one has My = Nx = . Since My and Nx equal, the equation is exact. Solve.
The ODE
[tex]M(x,y)\,\mathrm dx+N(x,y)\,\mathrm dy=0[/tex]
is exact if
[tex]\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}[/tex]
We have
[tex]M=-xy\sin x+2y\cos x\implies M_y=-x\sin x+2\cos x[/tex]
[tex]N=2x\cos x\implies N_x=2\cos x-2x\sin x[/tex]
so the ODE is indeed not exact.
Multiplying both sides of the ODE by [tex]\mu(x,y)=xy[/tex] gives
[tex]\mu M=-x^2y^2\sin x+2xy^2\cos x\implies(\mu M)_y=-2x^2y\sin x+4xy\cos x[/tex]
[tex]\mu N=2x^2y\cos x\implies(\mu N)_x=4xy\cos x-2x^2y\sin x[/tex]
so that [tex](\mu M)_y=(\mu N)_x[/tex], and the modified ODE is exact.
We're looking for a solution of the form
[tex]\Psi(x,y)=C[/tex]
so that by differentiation, we should have
[tex]\Psi_x\,\mathrm dx+\Psi_y\,\mathrm dy=0[/tex]
[tex]\implies\begin{cases}\Psi_x=\mu M\\\Psi_y=\mu N\end{cases}[/tex]
Integrating both sides of the second equation with respect to [tex]y[/tex] gives
[tex]\Psi_y=2x^2y\cos x\implies\Psi=x^2y^2\cos x+f(x)[/tex]
Differentiating both sides with respect to [tex]x[/tex] gives
[tex]\Psi_x=-x^2y^2\sin x+2xy^2\cos x=2xy^2\cos x-x^2y^2\sin x+\dfrac{\mathrm df}{\mathrm dx}[/tex]
[tex]\implies\dfrac{\mathrm df}{\mathrm dx}=0\implies f(x)=c[/tex]
for some constant [tex]c[/tex].
So the general solution to this ODE is
[tex]x^2y^2\cos x+c=C[/tex]
or simply
[tex]x^2y^2\cos x=C[/tex]
We are to verify and confirm if the given differential equations are exact or not. Then solve for the exact equation.
The first differential equation says:
[tex]\mathbf{(-xy \ sin x + 2y \ cos x) dx + 2(x \ cos x) dy = 0 }[/tex]
Recall that:
A differential equation that takes the form [tex]\mathbf{M(x,y)dt + N(x, y)dy = 0 }[/tex] will be exact if and only if:
[tex]\mathbf{\dfrac{\partial M }{\partial y} = \dfrac{\partial N }{\partial x}}[/tex]From equation (1), we can represent M and N as follows:
[tex]\mathbf{M = (-xy \ sin x + 2y \ cos x)}[/tex][tex]\mathbf{N = (2x \ cos x)}[/tex]Thus, taking the differential of M and N, we have:
[tex]\mathbf{ \dfrac{\partial M}{\partial y }= M_y = -x sin x + 2cos x}[/tex]
[tex]\mathbf{ \dfrac{\partial N}{\partial x }= N_x = 2 cos x + 2x sin x}[/tex]
From above, it is clear that:
[tex]\mathbf{\dfrac{\partial M }{\partial y} \neq \dfrac{\partial N }{\partial x}}[/tex]
∴
We can conclude that the equation is not exact.
Now, after multiplying the given differential equation in (1) by the integrating factor μ(x, y) = xy, we have:
[tex]\mathbf{ = \mathsf{(-x^2y^2 sin x + 2xy^2cos x ) dx +(2x^2ycos x ) dy = 0 --- (2)}}[/tex]Representing the equation into form M and N, then:
[tex]\mathbf{M = -x^22y^2 sin x +2xy^2 cos x}[/tex]
[tex]\mathbf{N = 2x^2y cos x}[/tex]
Taking the differential, we have:
[tex]\mathbf{\dfrac{\partial M}{\partial y }= M_y = -2x^2y sin x + 4xy cos x }[/tex]
[tex]\mathbf{\dfrac{\partial N}{\partial x} =N_x= 4xycos \ x -2x^2 y sin x}[/tex]
Here;
[tex]\mathbf{\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x} }[/tex]
Therefore, we can conclude that the second equation is exact.
Now, the solution of the second equation is as follows:
[tex]\int_{y } M dx + \int (not \ containing \ 'x') dy = C[/tex]
[tex]\rightarrow \int_{y } (-x^2y^2 sin(x) +2xy^2 cos (x) ) dx + \int(0)dy = C[/tex]
[tex]\rightarrow-y^2 \int x^2 sin(x) dx +2y ^2 \int x cos (x) dx = C[/tex] ---- (3)
Taking integrations by parts:
[tex]\int u v dx = u \int v dx - \int (\dfrac{du}{dx} \int v dx) dx[/tex]
∴
[tex]\int x^2 sin (x) dx = x^2 \int sin(x) dx - \int (\dfrac{d}{dx}(x^2) \int (sin \ (x)) dx) dx[/tex]
[tex]\to x^2 (-cos (x)) \ - \int 2x (-cos \ (x)) \ dx[/tex]
[tex]\to -x^2 (cos (x)) \ + \int 2x \ cos \ (x) \ dx[/tex] ----- replace this equation into (3)
∴
[tex]\rightarrow-y^2( -x^2 cos (x) \ + \int 2x \ cos \ (x) \ dx) +2y ^2 \int x cos (x) dx = C[/tex]
[tex]\mathbf{\rightarrow -x^2 y^2 cos (x) \ -2y ^2 \int x \ cos \ (x) \ dx +2y ^2 \int x cos (x) dx = C}[/tex]
[tex]\mathbf{x^2y^2 cos (x) = C\ \text{ where C is constant}}[/tex]
Therefore, from the explanation, we've can conclude that the first equation is not exact and the second equation is exact.
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Can someone plz help me and show your work I WILL MARK AS BRAINLIEST!!!! Plzzz someone!
By Pythagoras' Theorem:
Sum of the squares of the two side = Square of longest side
a² + b² = c²
a)
So let's check 7, 24, 25
Is 7² + 24² = 25² ?
7*7 + 24*24
49 + 576
=625.
Let us perform the other side 25²
25² = 25 * 25 = 625
Therefore the left hand side = Right hand side.
Therefore 7, 24, 25 is a Pythagorean Triple
b)
Let's check 9, 40, 41
Is 9² + 40² = 41² ?
9² + 40²
9*9+ 40*40
81 + 1600
=1681
Let us perform the other side 41²
41² = 41 * 41 = 1681
Therefore the left hand side = Right hand side.
Therefore 9, 40, 41 is a Pythagorean Triple.
Casie jumped off of a cliff into the ocean while on vacation. Her height as a function of time is modeled by the equation h = −16t2 +16t + 140, where t is the time in seconds and h is the height in feet. How long does it take Casie to hit the water?
A) 3 seconds
B) 3.5 seconds
C) 4 seconds
D) 4.5 seconds
Answer:
3.5 seconds, B
Step-by-step explanation:
This is an upside down parabola, a function that is extremetly useful in helping us to understand position and velocity and time and how they are all related. Her upwards velocity is 16 ft/sec and she starts from a height of 140 feet, according to the problem. The h is the height she ends up at after a certain amount of time has gone by. You want to know how long it will take her to hit the water. When she hits the water, she has no more height. Therefore, her height above the water when she hits the water is 0. Plug in a 0 for h and factor the quadratic to get t = -2.5 seconds and t = 3.5 seconds. The only two things in math that will never ever be negative is a distance measure and time, so we can disregard the -2.5 and go with 3.5 seconds as our answer.
Answer:
B
Step-by-step explanation:
There were 3 bananas, 4 apples, and 3 oranges in a basket. What is the probability that Ace will pick a banana from the basket?
0.3 or 30%. The probability that Ace pick a banana from a basket that content others fruits is 0.3.
The key to solve this problem is using the equation of probability [tex]P(A)=\frac{n(A)}{n}[/tex] where n(A) the numbers of favorables outcomes and n the numbers of possible outcomes.
There are in the basket 10 fruits in total (3 bananas + 4 apples + 3 oranges = 10fruits). Then, extract a fruit can occur in 10 ways, this is n. There is only 3 bananas in the basket, so the fruit that ACE will pick be a banana can occur in 3 ways out of 10, so 3 is n(A).
Solving the equation:
[tex]P(A)=\frac{3}{10}=0.3[/tex]
The probability that Ace will pick a banana from the basket is 3/10, as there are 3 bananas out of a total of 10 pieces of fruit.
The question asks for the probability that Ace will pick a banana from a basket containing 3 bananas, 4 apples, and 3 oranges. To calculate this, you sum up the total number of pieces of fruit, which is 3 bananas + 4 apples + 3 oranges = 10 pieces of fruit. The probability is then the number of desired outcomes (bananas) over the total number of possible outcomes (all pieces of fruit), which is 3 bananas / 10 pieces of fruit = 3/10 or 30%.
HELP!!!!!!!!!!!!! MAX POINTS
will you receive the grade immediately?
Answer:
21
Step-by-step explanation:
√(35x)
The prime factorization of 35 is 5×7. To simplify the radical, the expression underneath must be a multiple of a perfect square. So we need to choose a value of x that has either 5 or 7 as a factor.
21 has a factor of 7. Let's see:
√(35×21)
√(5×7×3×7)
√(15×7²)
7√15
The data table represents the distance between a well-known lighthouse and a cruise ship over time. The cruise ship is travelling at uniform speed. What will be the distance between the cruise ship and the lighthouse after 5 hours?
Number of Hours
Distance from Lighthouse (in oceanic miles)
2 53
4 95.5
6 138
8 180.5
10 223
12 265.5
14 308
16 350.5
84.50 oceanic miles
89.75 oceanic miles
116.75 oceanic miles
128.50 oceanic miles
223.00 oceanic miles
Answer:
116.75 oceanic miles
Step-by-step explanation:
A graph of the data shows the distance to be between 110 and 120 miles (closer to 120). There is only one answer choice in that range.
In 2 hours, the ship travels 42.5 miles, so in 1 hour will travel 21.25 miles. Adding that distance to the distance at 4 hours gives the distance at 5 hours, ...
95.5 +21.25 = 116.75 . . . . "oceanic" miles
_____
In order for the distance from the lighthouse to be uniformly increasing, the ship must be traveling directly away from the lighthouse. Traveling at any other angle, the distances will not fall on a straight line. (That is one reason I wanted to graph the data.)
Which of the following is not an equation of a simple, even polynomial function? y = | x | y = x2 y = x3 y = -x2
Answer:
y = | x |y = x^3Step-by-step explanation:
The absolute value function prevents the expression from being a polynomial. The degree of 3 in y^3 is an odd number so that polynomial function will not be even.
Answer:
The equation [tex]y=x^3[/tex] is not an equation of a simple , even polynomial function.
Step-by-step explanation:
Even function : A function is even when its graph is symmetric with respect to y-axis.
Algebrically , the function f is even if and only if
f(-x)=f(x) for all x in the domain of f.
When the function does not satisfied the above condition then the function is called non even function.
f(x)[tex]\neq[/tex] f(-x)
Now , we check given function is even or not
A. y= [tex]\mid x\mid[/tex]
If x is replaced by -x
Then we get the function
f(-x)=[tex]\mid -x \mid[/tex]
f(-x)=[tex]\mid x \mid[/tex]
Hence, f(-x)=f(x)
Therefore , it is even polynomial function.
B. [tex]y=x^2[/tex]
If x is replace by -x
Then we get
f(-x)=[tex](-x)^2[/tex]
f(-x)=[tex]x^2[/tex]
Hence, f(-x)=f(x)
Therefore, it is even polynomial function.
C. [tex]y=x^3[/tex]
If x is replace by -x
Then we get
f(-x)=[tex](-x)^3[/tex]
f(-x)=[tex]-x^3[/tex]
Hence, f(-x)[tex]\neq[/tex] f(x)
Therefore, it is not even polynomial function.
D.[tex]y= -x^2[/tex]
If x is replace by -x
Then we get
f(-x)= - [tex](-x)^2[/tex]
f(-x)=-[tex]x^2[/tex]
Hence, f(-x)=f(x)
Therefore, it is even polynomial function.
Answer: C. [tex]y=x^3[/tex] is not simple , even polynomial function.
The function f(x) = 0.11x + 43 relates how much Derek pays for phone service, f(x), to the number of minutes, x, used for international calls in a month. What is the value and meaning of f(320)?
Explanation:
To find the value, put 320 where x is and do the arithmetic.
f(320) = 0.11·320 +43 = 35.20 +43 = 78.20
The meaning is described by the problem statement:
"how much Derek pays for phone service" for "the number of minutes, [320], used for international calls in a month."
Derek pays 78.20 for 320 minutes of international calls in a month.
__
The units (dollars, rupees, euros, pounds, ...) are not specified.
Answer:
Given the function f(x) = 0.11x + 43, this shows the relationship between how much Derek has to pay for phone service for the amount of minutes he uses on international calls a month. f(320) can be solved by substituting x = 320, and this is shown below: f(x) = 0.11x + 43 f(320) = 0.11(320) + 43 f(320) = 78.2 This means that Derek has to pay $78.20 for the 320 minutes of calls. Among the choices, the correct answer is B.
Given: K=2∙33∙11∙172and M=3∙11∙173
Evaluate 18·M÷K.
Answer:
1557/1892
Step-by-step explanation:
Your calculator can do this:
[tex]\dfrac{18M}{K}=\dfrac{18\cdot 3\cdot 11\cdot 173}{2\cdot 33\cdot 11\cdot 172}=\dfrac{18\cdot 173}{2\cdot 11\cdot 172}\\\\=\dfrac{1557}{1892}[/tex]
Given: LMNB is a square, LM = 20cm, P∈ LM , K ∈ PN , PK = 1 5 PN, LP = 4 cm Find: Area of LPKB
Answer:
80 cm²
Step-by-step explanation:
Trapezoid LPKB has area ...
A = (1/2)(b1 +b2)h = (1/2)(4 +20)(20) = 240 . . . . cm²
Triangle BPN has area ...
A = (1/2)bh = (1/2)(20)(20) = 200 . . . . cm²
Triangle BKN has a height that is 4/5 the height of triangle BPN, so will have 4/5 the area:
ΔBKN = (4/5)(200 cm²) = 160 cm²
The area of quadrilateral LPKB is that of trapezoid LPNB less the area of triangle BKN, so is ...
240 cm² - 160 cm² = 80 cm²
What is the value when c =6 and d= 10 5c2 - 3d + 15
Answer:
165
Step-by-step explanation:
[tex]5c^{2} -3d+15[/tex]
c = 6 and d = 10
[tex]5c^{2}[/tex] = 5 × 6² = 5 × 36 = 180
[tex]5c^{2}[/tex] - ( 3 d ) = 180 - ( 3 × 10 ) = 180 - 30 = 150
[tex]5c^{2}[/tex] - 3 d ( + 15 ) = 150 + 15 = 165
Answer:
165
Step-by-step explanation:
Substitutet 6 for c and 10 for d in 5c^2 - 3d + 15 .
Note that " ^ " is used here to denote exponentiation; c2 is meaningless.
Then we have 5(6)^2 - 3(10) + 15, or 180 - 30 + 15, or 165.
105. Suppose that the probability that an adult in America will watch the Super Bowl is 40%. Each person is considered independent. We are interested in the number of adults in America we must survey until we find one who will watch the Super Bowl. a. In words, define the random variable X. b. List the values that X may take on. c. Give the distribution of X. X ~ _____(_____,_____) d. How many adults in America do you expect to survey until you find one who will watch the Super Bowl? e. Find the probability that you must ask seven people. f. Find the probability that you must ask three or four people.
Answer:
a. X is the number of adults in America that need to be surveyed until finding the first one that will watch the Super Bowl.
b. X can take any integer that is greater than or equal to 1. [tex]\rm X\in \mathbb{Z}^{+}[/tex].
c. [tex]\rm X \sim NB(1, 0.40)[/tex].
d. [tex]E(\rm X) = 2.5[/tex].
e. [tex]P(\rm X = 7) = 0.0187[/tex].
f. [tex]P(\text{X} = 3) +P(\text{X} = 4) = 0.230[/tex].
Step-by-step explanation:
a.In this setting, finding an adult in America that will watch the Super Bowl is a success. The question assumes that the chance of success is constant for each trial. The question is interested in the number of trials before the first success. Let X be the number of adults in America that needs to be surveyed until finding the first one who will watch the Super Bowl.
b.It takes at least one trial to find the first success. However, there's rare opportunity that it might take infinitely many trials. Thus, X may take any integer value that is greater than or equal to one. In other words, X can be any positive integer: [tex]\rm X\in \mathbb{Z}^{+}[/tex].
c.There are two discrete distributions that may model X:
The geometric distribution. A geometric random variable measures the number of trials before the first success. This distribution takes only one parameter: the chance of success on each trial. The negative binomial distribution. A negative binomial random variable measures the number of trials before the r-th success. This distribution takes two parameters: the number of successes [tex]r[/tex] and the chance of success on each trial [tex]p[/tex].[tex]\rm NB(1, p)[/tex] (note that [tex]r=1[/tex]) is equivalent to [tex]\sim Geo(p)[/tex]. However, in this question the distribution of [tex]\rm X[/tex] takes two parameters, which implies that [tex]\rm X[/tex] shall follow the negative binomial distribution rather than the geometric distribution. The probability of success on each trial is [tex]40\% = 0.40[/tex].
[tex]\rm X\sim NB(1, 0.40)[/tex].
d.The expected value of a negative binomial random variable is equal to the number of required successes over the chance of success on each trial. In other words,
[tex]\displaystyle E(\text{X}) = \frac{r}{p} = \frac{1}{0.40} = 2.5[/tex].
e.[tex]P(\rm X = 7) = 0.0187[/tex].
Some calculators do not come with support for the negative binomial distribution. There's a walkaround for that as long as the calculator supports the binomial distribution. The r-th success occurs on the n-th trial translates to (r-1) successes on the first (n-1) trials, plus another success on the n-th trial. Find the chance of (r-1) successes in the first (n-1) trials and multiply that with the chance of success on the n-th trial.
f.[tex]P(\text{X} = 3)+P(\text{X} = 4) = 0.230 [/tex].
Solve this gear problem.
Gear 1 = 30 teeth
Speed, gear 1 = 150 r.p.m.
Speed, gear 2 = 50 r.p.m.
Teeth, gear 2 = ?
The answer is:
The number of teeth of Gear 2 is 90 teeth.
[tex]N_{2}=90teeth[/tex]
Why?To calculate the number of teeth for the Gear 2, we need to use the following formula that establishes a relation between the number of RPM and the number of teeth of two or more gears.
[tex]N_{1}Z_{1}=N_{2}Z_{2}[/tex]
Where,
N, are the rpm of the gears
Z, are the teeth of the gears.
We are given the following information:
[tex]Z_{1}=30teeth\\N_{1}=150RPM\\N_{2}=50RPM[/tex]
Then, substituting and calculating we have:
[tex]N_{1}Z_{1}=N_{2}Z_{2}[/tex]
[tex]150RPM*30teeth=N_{2}50RPM[/tex]
[tex]N_{2}=\frac{150RPM*30teeth}{50RPM}=90teeth[/tex]
[tex]N_{2}=90teeth[/tex]
Hence, we have that the number of teeth of Gear 2 is 90 teeth.
Have a nice day!
Eric, George, and Denzel have invested $400,000, $300,000, and $300,000, respectively, in a business venture. They have decided that they will divide the profits among themselves in the ratio of their respective investments. If their business makes a profit of $75,000, what would be Eric’s share in the profit? A. $22,500 B. $30,000 C. $32,500 D. $45,000
Answer:
Eric’s share in the profit is $30,000 ⇒ answer B
Step-by-step explanation:
* We will use the ratio to solve this problem
- At first lets find the ratio between their invested
∵ Eric has invested $400,000
∵ George has invested $300,000
∵ Denzel has invested $300.000
- To find the ratio divide each number by 100,000
∴ Eric : George : Denzel = 4 : 3 : 3
- They will divide the profits among themselves in the ratio of their
respective investments
- The total profit will divided by the total of their ratios
∵ The total of the ratios = 4 + 3 + 3 = 10
∴ Eric : George : Denzel : Sum = 4 : 3 : 3 : 10
- That means the profit will divided into 10 equal parts
- Eric will take 4 parts, George will take 3 parts and Denzel will take
3 parts
∵ The profit = $75,000
- Divide the profit by the sum of the ratio
∴ Each part of the profit = 75,000 ÷ 10 = $7,500
- Now lets find the share of each one
∴ The share of Eric = 4 × 7,500 = $30,000
∴ The share of George = 3 × 7,500 = $22,500
∴ The share of Denzel = 3 × 7,500 = $22,500
* Eric’s share in the profit is $30,000
# If you want to check your answer add the shares of them, the answer
will be the total profit (30,000 + 22,500 + 22,500 = $75,000), and if
you find the ratio between their shares it will be equal the ratio
between their investments (divide each share by 7,500 to simplify
them the answer will be 4 : 3 : 3)
Answer:
B
Step-by-step explanation:
rowan wants to justify that f(x) 3x-7 is a linear function. If she evaluates f(x) for consecutive integer values, which statement justices the claim that f is a linear function?
a. there is a common difference of -7 for f(x) when x increase by 1
b. there is a common factor of -7 for f(x) when x increase by 1
c. there is a common different of 3 for f(x) when x increase by 1
d. there is a common factor of 3 for f(x) when x increase by 1
Answer:
C. there is a common difference of 3 for f(x) when x increases by 1
Step-by-step explanation:
As 3 is the slope of this function, there will be a common difference of 3 when x increases by 1.
f(x) = 3x - 7
Let's think, whenever we add 1 to x it i'll increase 3 in the result
f(0) = 3.0 - 7 = 0 - 7 = -7
f(1) = 3.1 - 7 = 3 - 7 = -4
f(2) = 3.2 - 7 = 6 - 7 = -1
So we can know that there's a common difference of 3 for f(x) when x increase by 1.